Protein charge transfer far from equilibrium: a theoretical perspective

Potential differences for protein-assisted electron transfer across lipid bilayers or in bio-nano setups can amount to several 100 mV; they lie far outside the range of linear response theory. We describe these situations by Pauli-master equations that are based on Marcus theory of charge transfer between self-trapped electrons and that obey Kirchhoff's current law. In addition, we take on-site blockade effects and a full non-linear response of the local potentials into account. We present analytical and numerical current-potential curves and electron populations for multi-site model systems and biological electron transfer chains. Based on these, we provide empirical rules for electron populations and chemical potentials along the chain. The Pauli-master mean-field results are validated by kinetic Monte Carlo simulations. We briefly discuss the biochemical and evolutionary aspects of our findings.

The Unified Redox Scale for All Solvents: Consistency and Gibbs Transfer Energies of Electrolytes from their Constituent Single Ions

We have devised the unified redox scale Eabs H2O , which is valid for all solvents. The necessary single ion Gibbs transfer energy between two different solvents, which only can be determined with extra-thermodynamic assumptions so far, must clearly satisfy two essential conditions: First, the sum of the independent cation and anion values must give the Gibbs transfer energy of the salt they form. The latter is an observable and measurable without extra-thermodynamic assumptions. Second, the values must be consistent for different solvent combinations. With this work, potentiometric measurements on silver ions and on chloride ions show that both conditions are fulfilled using a salt bridge filled with the ionic liquid [N2225 ][NTf2 ]: if compared to the values resulting from known pKL values, the silver and chloride single ion magnitudes combine within a uncertainty of 1.5 kJ mol-1 to the directly measurable transfer magnitudes of the salt AgCl from water to the solvents acetonitrile, propylene carbonate, dimethylformamide, ethanol, and methanol. The resulting values are used to further develop the consistent unified redox potential scale Eabs H2O that now allows to assess and compare redox potentials in and over six different solvents. We elaborate on its implications.

Measurements and Utilization of Consistent Gibbs Energies of Transfer of Single Ions: Towards a Unified Redox Potential Scale for All Solvents

Utilizing the "ideal" ionic liquid salt bridge to measure Gibbs energies of transfer of silver ions between the solvents water, acetonitrile, propylene carbonate and dimethylformamide results in a consistent data set with a precision of 0.6 kJ mol-1 over 87 measurements in 10 half-cells. This forms the basis for a coherent experimental thermodynamic framework of ion solvation chemistry. In addition, we define the solvent independent pe abs H 2 O - and the E abs H 2 O values that account for the electronating potential of any redox system similar to the pH abs H 2 O value of a medium that accounts for its protonating potential. This E abs H 2 O scale is thermodynamically well-defined enabling a straightforward comparison of the redox potentials (reducities) of all media with respect to the aqueous redox potential scale, hence unifying all conventional solvents' redox potential scales. Thus, using the Gibbs energy of transfer of the silver ion published herein, one can convert and unify all hitherto published redox potentials measured, for example, against ferrocene, to the E abs H 2 O scale.

Monte Carlo simulation and thermodynamic integration applied to protein charge transfer

We introduce a combination of Monte Carlo simulation and thermodynamic integration methods to address a model problem in free energy computations, electron transfer in proteins. The feasibility of this approach is tested using the ferredoxin protein from Clostridium acidurici. The results are compared to numerical solutions of the Poisson-Boltzmann equation and data from recent molecular dynamics simulations on charge transfer in a protein complex, the NrfHA nitrite reductase of Desulfovibrio vulgaris. Despite the conceptual and computational simplicity of the Monte Carlo approach, the data agree well with those obtained by other methods. A link to experiments is established via the cytochrome subunit of the bacterial photosynthetic reaction center of Rhodopseudomonas viridis.

Long-range electron-electron interaction and charge transfer in protein complexes: a numerical approach

With application to the nitrite reductase hexameric protein complex of Desulfovibrio vulgaris, NrfH2A4, we suggest a strategy to compute the energy landscape of electron transfer in large systems of biochemical interest. For small complexes, the energy of all electronic configurations can be scanned completely on the level of a numerical solution of the Poisson-Boltzmann equation. In contrast, larger systems have to be treated using a pair approximation, which is verified here. Effective Coulomb interactions between neighbouring sites of excess electron localization may become as large as 200 meV, and they depend in a nontrivial manner on the intersite distance. We discuss the implications of strong Coulomb interactions on the thermodynamics and kinetics of charging and decharging a protein complex. Finally, we turn to the effect of embedding the system into a biomembrane.

Charge transfer through a fragment of the respiratory complex I and its regulation: an atomistic simulation approach

We simulate electron transfer within a fragment of the NADH:ubiquinone oxidoreductase (respiratory complex I) of the hyperthermophilic bacterium Aquifex aeolicus. We apply molecular dynamics simulations, thermodynamic integration, and a thermodynamic network least squares analysis to compute two key parameters of Marcus' theory of charge transfer, the thermodynamic driving force and the reorganization energy. Intramolecular contributions to the Gibbs free energy differences of electron and hydrogen transfer processes, ΔG, are accessed by calibrating against experimental redox titration data. This approach permits the computation of the interactions between the species NAD+, FMNH2, N1a-, and N3-, and the construction of a free energy surface for the flow of electrons within the fragment. We find NAD+ to be a strong candidate for the regulation of charge transfer.

Simulating biological charge transfer: Continuum dielectric theory or molecular dynamics?

We study the thermodynamic parameters of Marcus's theory of charge transfer, the driving forces and the reorganization energies, using two widely applied approaches to bioenergetic problems that seem to be radically different: continuum dielectric theory via a numerical solution of Poisson's equation, and the thermodynamic integration approach based upon classical Newtonian molecular dynamics, as perfomed by Na et al., PCCP 19, 18,938 (2017). With application to a nitrite reductase NrfHA protein heterodimer, we obtain an excellent agreement between the respective driving forces with an r.m.s. deviation of 1.7 kcal/mol, and a lower limit to the reorganization energies. The computational methods turn out to be mutually supportive: molecular dynamics can be used to determine the parameters of a dielectric theory computation, which on the other hand can be used to properly rescale the reorganization energies and partition them into aqueous and protein contributions. In addition, we use the electrostatic approach to study the influence of Ca²⁺ ions on the free energy landscape of charge transfer.

Understanding the hydrogen transfer mechanism for the biodegradation of 2,4,6-trinitrotoluene catalyzed by pentaerythritol tetranitrate reductase: molecular dynamics simulations

The explosive 2,4,6-trinitrotoluene (TNT) is a highly toxic pollutant.